Differentiable Manifolds, Tensors, and Curvature

The differential geometry of curves and surfaces, and its generalisation to differentiable manifolds of arbitrary dimension, provides the language in which modern physics describes configuration spaces, spacetimes, and gauge fields. This chapter builds that language in the order in which the source manuscript develops it: tensors under orthogonal transformations in Euclidean space; the classical theory of curves and surfaces in \(\R^{3}\), where the metric, the Christoffel symbols, and the Riemann symbols first appear in concrete form; the definition of a differentiable manifold through charts and atlases; tensor analysis on manifolds and under general coordinate transformations; differential forms; the metric tensor; the covariant derivative and affine connections; and the curvature tensors.

The chapters on general relativity build directly on this material [Wald:1984] [Misner:1973].

Index conventions and the symmetry ladder

This chapter, and every chapter that builds on it, works in general dimension \(D = p + q\) with metric signature \((p,q)\) (project rule, Epistemology and the Scientific Method); the observed case is the instantiation \(p + q = 3 + 1\) of Part IV — Special Relativity and Part V — General Relativity and Cosmology. Two families of transformations act on everything we construct: diffeomorphisms (general coordinate transformations of the manifold) and local frame transformations (rotations of the orthonormal frames introduced in Section 13.8). Moreover, the maximally symmetric geometries of Section 13.13 are most transparently described by embedding the \((p+q)\)-dimensional space in flat spaces of one and two dimensions more. We therefore fix, once and for all, the covariant index notation for tensors and connections defined in \((p+q)\)-, \((p+q+1)\)- and \((p+q+2)\)-dimensional spaces. The following types of letters are used for indices of (pseudo-)tensors and connections under diffeomorphisms:

while for the indices of (pseudo-)tensors and connections under local Lorentz, (anti-)de Sitter and conformal transformations we use

respectively.

The groups these indices transform under form a ladder that recurs throughout the treatise; we fix its names and symbols in Table 13.1.

The frame metric of signature \((p,q)\) is fixed as

and likewise \(\eta_{AB}\) and \(\eta_{IJ}\) denote the flat metrics of the signatures indicated by the corresponding group in Table 13.1 — \((p,q+1)\) for \(\operatorname{AdS}_{p+q}\), \((p+1,q)\) for \(\operatorname{dS}_{p+q}\), and \((p+1,q+1)\) for \(\operatorname{C}_{p+q}\). In the relativistic chapters of the physical parts, where \(p+q = 3+1\) and the signature convention is \((-,+,+,+)\) (Part IV — Special Relativity), the dictionary is \((p,q) = (3,1)\) with the roles of the signs read from context; nothing in this chapter depends on the overall sign choice.

Tensors under orthogonal transformations

Before tensors are defined on a general manifold, it is instructive to meet them in their simplest habitat: Euclidean space referred to orthogonal coordinate systems, where all indices can be written as subscripts and the transformation matrices are constant.

Orthogonal coordinate transformations

Linear transformations

Differentiating Equation (13.2) we see that

From Equations (13.2) and (13.3) we have

Renaming indices, we obtain the inverse transformation of the coordinates,

We denote

which is justified because

where \(\left(\Lambda^{-1}\right)_{ik}\) is the inverse transformation matrix, with constant coefficients. Therefore, from Equation (13.4),

Orthogonal coordinate systems

Owing to Equation (13.8), the unit vectors of the new basis are related to the old ones by

and since the new coordinate system is also orthogonal,

Geometric interpretation of the transformation matrix

From Equations (13.7) and (13.9) we see that

On the other hand, let \(\theta_{ij}\) be the angles between the unit vectors \(\hat{e}_{j}\) and \(\hat{e}'_{i}\). Then

so that, by Equation (13.11),

The transformation matrix therefore contains the cosines of the angles formed between the unit vectors \(\hat{e}_{j}\) and \(\hat{e}'_{i}\).

Orthogonality condition

Consider two orthogonal coordinate systems \(K\) and \(K'\) related by an orthogonal transformation of the form Equation (13.8). Since \(K'\) is orthogonal, from Equations (13.7), (13.9) and (13.10),

Hence the orthogonality condition reads

In matrix notation this corresponds to

Taking the determinant of both sides,

Since \(\det(A)\neq 0\), the inverse \(A^{-1}\) exists, with

Comparing Equation (13.14) with Equation (13.16) we see that for orthogonal transformations

Since Equation (13.16) may equally be written \(A^{-1}A = I\), from Equation (13.17) we also have \(A\transpose A = I\); in index notation the orthogonality condition can therefore also be written

Inverse orthogonal transformation

From Equations (13.8) and (13.18),

Thus the inverse orthogonal coordinate transformation is given by

Finally, differentiating Equation (13.8) with respect to \(x_{j}\) and Equation (13.19) with respect to \(x'_{i}\), we see that

Cartesian tensors

In particular, for \(r = 2\),

To find the inverse relation, note from Equations (13.18) and (13.21) that

so that

Notable Cartesian tensors

We call vector a tensor of rank \(1\).

From the orthogonality condition Equation (13.13) we may write

Moreover, by its definition the Kronecker delta takes the same values in every orthogonal coordinate system, so we may write

Comparing with Equation (13.22), the Kronecker delta satisfies the transformation law of a rank-\(2\) tensor. Therefore the Kronecker delta is an invariant tensor under orthogonal transformations.

Algebra of Cartesian tensors

Addition

Consider the quantities \(A_{i_1\cdots i_r}\) and \(B_{i_1\cdots i_r}\), components of tensors under orthogonal transformations, both of rank \(r\). Define the quantities

Then, by Equation (13.21),

Therefore the addition of tensors forms the components of a new tensor under orthogonal transformations.

Multiplication by a scalar

Consider the quantities \(A_{i_1\cdots i_r}\), components of a rank-\(r\) tensor under orthogonal transformations, and a scalar \(\lambda\). Define

Then

Therefore the multiplication of a tensor by a scalar forms the components of a new tensor under orthogonal transformations.

Product

Consider the quantities \(A_{i_1\cdots i_r}\) and \(B_{i_1\cdots i_s}\), components of tensors under orthogonal transformations, of ranks \(r\) and \(s\) respectively. Define

Then, by Equation (13.21),

Therefore the product of tensors forms the components of a new tensor under orthogonal transformations.

Permutation of indices

Consider the quantities \(A_{i_1\cdots i_r}\), components of a rank-\(r\) tensor under orthogonal transformations. Define the symbol

in which the \(s\)-th and \(t\)-th indices of \(A_{i_1\cdots i_r}\) are permuted. Then, by Equation (13.21) and reordering the constant factors \(a_{ij}\),

Thus the permutation of indices of a tensor defines a new tensor under orthogonal transformations.

Contraction of indices

Consider the quantities \(A_{i_1\cdots i_r}\), components of a rank-\(r\) tensor under orthogonal transformations. Define the \(n^{r-2}\) quantities

Then, by Equations (13.18) and (13.21),

Thus the contraction of two indices of a tensor defines a new tensor, of rank \(r-2\), under orthogonal transformations.

Quotient law

Consider the quantities \(A_{i_1\cdots i_sj_1\cdots j_r}\), and consider further the quantities \(B_{j_1\cdots j_r}\), components of a rank-\(r\) tensor under orthogonal transformations. Let \(C_{i_1\cdots i_s}\) be given by

With respect to another orthogonal coordinate system,

Supposing that the \(C_{i_1\cdots i_s}\) are the components of a rank-\(s\) tensor under orthogonal transformations, from Equation (13.21) we have

Since the \(B_{k_1\cdots k_r}\) are arbitrary, applying \(a_{m_1k_1}\cdots a_{m_rk_r}\) to both sides of the resulting relation and using Equation (13.13),

Hence the quantities \(A_{i_1\cdots i_sj_1\cdots j_r}\) satisfying Equation (13.28) are the components of a tensor under orthogonal transformations. This result is called the quotient law.

The Levi-Civita symbol

Beside the Kronecker delta, three dimensions carry a second array with the same components in every frame. Define \(\varepsilon_{ijk}\), with \(i,j,k\in\set{1,2,3}\), to be totally antisymmetric with \(\varepsilon_{123}=+1\); equivalently \(\varepsilon_{ijk}=\sgn\sigma\) when \((i,j,k)=(\sigma(1),\sigma(2),\sigma(3))\) for a permutation \(\sigma\), and zero when two indices coincide. (The general \(n\)-index symbol reappears in Section 13.5.3 as a density of weight \(-1\) under arbitrary coordinate changes; under orthogonal changes the weight factor is \(\det a=\pm1\), which is the content of Equation (13.31) below.)

Isotropic tensors

The algebra above says how tensors combine. The question left open is which tensors are singled out by the group itself — those whose components are the same in every frame. That question is invariant theory for the rotation group rather than tensor algebra, and it is settled once and for all in Theorem 5.128. In three dimensions — and the dimension is essential, as Remark 13.7 records — a Cartesian tensor (Definition 13.5) whose components are unchanged by every proper rotation vanishes at rank one, is a multiple of \(\delta_{ij}\) at rank two, a multiple of \(\varepsilon_{ijk}\) at rank three, and a combination

of the three products of Kronecker deltas at rank four — three free numbers in place of \(3^{4}=81\). The invariance of \(\delta_{ij}\) and of \(\varepsilon_{ijk}\) under proper rotations, which is what makes the converse half true, is Equations (13.24) and (13.31) of this section.

The theory of curves and surfaces in $\R^{3}$

Theory of curves

Curves in $\R^{3}$

From this we see that the inverse function of \(s\) is an admissible change of parameter for \(t\). Moreover, differentiating both sides of Equation (13.35) with respect to \(t\),

and therefore, from Equation (13.33),

From Equation (13.37) we see that \(\hat{T}\) is indeed a unit vector.

The derivative of any unit vector \(\hat{e}\) lies in the plane orthogonal to \(\hat{e}\), for

and therefore the vector \(\vect{k}\) lies in the plane orthogonal to \(\hat{T}\).

We define the curvature function \(\kappa\) as

We thus find a relation between the derivative of the unit tangent vector and the unit normal,

We thereby obtain a system of three orthonormal vectors called the moving trihedron (the Frenet frame).

Torsion

Differentiating Equation (13.44) with respect to \(s\),

From Equation (13.43) the first term of this expression vanishes. From Equation (13.40) the vector \(\dv{\hat{N}}{s}\) lies in the plane spanned by \(\hat{T}\) and \(\hat{B}\), so we may write \(\dv{\hat{N}}{s}\) as a linear combination of \(\hat{T}\) and \(\hat{B}\). Thus

where the function \(\tau\) is called the torsion of the curve.

From Equation (13.44) we see that

and by Equations (13.43) and (13.46),

The Frenet–Serret equations

Equations Equations (13.43), (13.46) and (13.47) are the Frenet–Serret equations, which relate the vectors of the moving trihedron to their derivatives:

or, in matrix form,

The fundamental theorem of the theory of curves

The Frenet–Serret system determines the moving trihedron from the two functions \(\kappa\) and \(\tau\), and the trihedron determines the curve. The precise statement is the following.

Theory of surfaces

Surfaces in $\R^{3}$

From the definition we see that these curves lie on the surface.

The tangent vectors to the curves of parameter \(u_1\) and \(u_2\) are given by

respectively. These vectors lie in the tangent plane of the surface.

First fundamental form

Consider the differential

This differential is written as a linear combination of the vectors \(\vect{x}_{1}\) and \(\vect{x}_{2}\), and therefore also lies in the tangent plane of the surface. Consider a curve on the surface, given by

where \(s\) is the arc-length parameter of the curve. From Equation (13.36) we have

where \(\dd s\) is called the arc element. Thus

From Equations (13.55) and (13.58) we see that

The first fundamental form is therefore given by

From the definition we see that the \(g_{ij}\) are the components of a symmetric tensor of rank \(2\), which is called the metric tensor, or simply the metric. The determinant of the metric tensor is given by

We can compute these quantities using matrix notation: writing

with

we see from Equation (13.64) that \([g^{ij}] = [g_{li}]^{-1}\), and therefore, from the algebra of matrices,

Length of a curve and area of a surface

Consider the curve on a surface \(\vect{x}\) given by Equation (13.56). Since the surface carries a first fundamental form, we can compute the length of the whole curve in terms of the first fundamental coefficients. As \(s = \int\dd s\), the length of the whole curve is given by

Now define an angle \(\gamma\) between the vectors \(\vect{x}_{1}\) and \(\vect{x}_{2}\). Since

from Equation (13.60) we have

On the other hand, the area of an infinitesimal parallelogram of the surface is given by

and from Equation (13.69),

Thus we see that

Integrating, the area of a surface is

Second fundamental form

The differential of any unit vector \(\hat{e}\) lies in the plane orthogonal to \(\hat{e}\), for

so the vector \(\dd\hat{n}\) lies in the plane orthogonal to \(\hat{n}\). By the chain rule, the differential \(\dd\hat{n}\) is given by

From Equations (13.55) and (13.74) we see that

Thus the second fundamental form is given by

From their definition, the second fundamental coefficients are the components of a symmetric tensor of rank \(2\). The determinant of the matrix formed by the second fundamental coefficients is given by

Alternative expression.

Note that the unit normal \(\hat{n}\) is orthogonal to each of the partial tangent vectors, \(\hat{n}\cdot\vect{x}_{i} = 0\). Hence

and therefore, from Equation (13.78), the second fundamental form is given by

Geometric interpretation.

Recall from linear algebra that the projection \(\vect{P}\) of a vector \(\vect{A}\) onto another vector \(\vect{B}\) is given by

Consider now the vector that, with respect to a tangent plane of the surface, indicates the small advance of the surface in a neighbourhood of the point of tangency. This vector is \(\vect{x}(u_1+\dd u_1,u_2+\dd u_2)-\vect{x}(u_1,u_2)\); thus the distance \(d\) from the surface to its tangent plane, in a neighbourhood of the point of tangency, equals the projection of this vector onto the unit normal \(\hat{n}\). From Equation (13.82),

Making a multivariable Taylor expansion of \(\vect{x}(u_1+\dd u_1,u_2+\dd u_2)\) about \((u_1,u_2)\),

Therefore the second fundamental form is, approximately, twice the distance from the surface to its tangent plane in a neighbourhood of the point of tangency.

The Gauss equations and the Christoffel symbols

Since at each point of the surface we have the three linearly independent vectors \(\vect{x}_{1}\), \(\vect{x}_{2}\), and \(\hat{n}\), we can write any vector of \(\R^{3}\) as a linear combination of them. In particular, we can express the partial derivative of \(\vect{x}_{i}\) with respect to \(u_{j}\) as

where \(\Gamma_{ij}^{\ k}\) and \(\alpha_{ij}\) are the coefficients of the linear combination. Taking the scalar product with \(\hat{n}\) and using Equation (13.80),

Thus we obtain the Gauss equations,

From Equation (13.83), taking the scalar product with \(\vect{x}_{l}\) and using Equations (13.54) and (13.60),

Note from their definition that the Christoffel symbols are symmetric in their first two indices: considering that \(\vect{x}\) is a map of class \(C^{2}\), by Schwarz's theorem

Alternative expression for the Christoffel symbols.

From the definition of the first fundamental coefficients, Equation (13.60),

In the same way,

Then, since the Christoffel symbols are symmetric in their first two indices,

and therefore

From Equation (13.86), multiplying Equation (13.87) by \(g^{kl}\) and renaming indices,

Note from Equation (13.88) that setting \(j = k\) gives

where the last two terms cancel after renaming the summed indices; therefore

The Weingarten equations

By the same justification as for the Gauss equations, we can write the partial derivative of \(\hat{n}\) with respect to \(u_{i}\) as a linear combination of the vectors \(\vect{x}_{1}\), \(\vect{x}_{2}\), and \(\hat{n}\). Moreover, in analogy with Equations (13.40) and (13.73), the partial derivative of any unit vector \(\hat{e}\) with respect to \(u_{i}\) lies in the plane orthogonal to \(\hat{e}\):

Then \(\hat{n}_{i}\) lies in the plane orthogonal to \(\hat{n}\), so we may write

where the quantities \(\beta_{i}^{\ j}\) are the factors of the linear combination. To compute them, take the scalar product with \(\vect{x}_{k}\) and use Equations (13.60) and (13.77):

Finally, we obtain the Weingarten equations,

The Gauss–Codazzi equations

Starting from the Gauss equations,

Renaming indices, we have

By the compatibility conditions, \(\vect{x}_{ijk}-\vect{x}_{ikj}=\vect{0}\); subtracting the previous equations,

Since \(\vect{x}_{l}\) and \(\hat{n}\) are linearly independent, their factors vanish separately. From this we see that

which are the Gauss–Codazzi equations. These equations relate the metric to the second fundamental coefficients.

The Riemann symbols

Curvatures of a surface

Normal curvature

Let \(\vect{x}=\vect{x}(u_1(t),u_2(t))\) be a curve whose points lie on the surface \(\vect{x}=\vect{x}(u_1,u_2)\), and consider the normal curvature vector \(\vect{k}\) of that curve (see Equation (13.39)).

We call normal curvature the quantity

Let us derive an alternative expression for the normal curvature. We have, using Equations (13.37) and (13.39),

Since the tangent vector \(\hat{T}\) of the curve is orthogonal to the surface normal \(\hat{n}\),

Therefore Equation (13.99) becomes, using Equations (13.37) and (13.38),

Finally, we arrive at

Principal curvatures and Gaussian curvature

From Equation (13.100) we see that

Introducing the auxiliary variable

we have

We now seek an extremal value of \(k_{n}\), denoted \(k_{np}\), which we call a principal curvature. That is,

This relation holds for a value of \(\lambda\) that makes \(k_{n}\) extremal. Substituting it back into Equation (13.103),

Note that Equation (13.104) can be rewritten in the form

and substituting this result into Equation (13.105) yields a second expression for the principal curvature,

Returning to the original variables, Equations (13.105) and (13.106) give the system of equations

Being a homogeneous system of two equations in the two variables \(\dd u_{1}\) and \(\dd u_{2}\), it has a non-trivial solution if and only if the determinant of the system vanishes:

We thus obtain a quadratic equation for \(k_{np}\) in terms of the first and second fundamental coefficients. Recall from the algebra of polynomials that for a quadratic equation \(ax^{2}+bx+c=0\) the product of its roots \(x_{1}\) and \(x_{2}\) is \(x_{1}x_{2}=c/a\).

Geometric interpretation of the Gaussian curvature

The sign of \(K\) classifies the local shape of the surface at a point: where \(K>0\) the two principal curvatures share a sign and the surface curves away from its tangent plane on every side (elliptic point, as on a sphere or an ellipsoid); where \(K<0\) they have opposite signs and the surface crosses its tangent plane (a saddle, as on a hyperboloid or the inner rim of a torus); where \(K=0\) at least one principal curvature vanishes (a cylinder, a cone, or a plane). Figure 13.1 illustrates the flat and cylindrical cases, both with \(K=0\).

Theorema Egregium

The definition \(K=b/g\) (Equation (13.107)) is built from both fundamental forms: \(g_{ij}\), intrinsic to the surface, and \(b_{ij}\), which records how the surface bends inside the ambient space. Gauss's celebrated theorem is that the second form is, in the end, dispensable.

A flat sheet of paper bent, without stretching or tearing, into a cylinder realizes exactly this corollary: the bending is a local isometry (distances measured along the surface do not change), and indeed both surfaces have \(K=0\) (Figure 13.1). The same sheet cannot be bent into any piece of a sphere of radius \(R\), whose curvature \(K=1/R^{2}\neq0\) (Example 13.33 below) can never match the flat sheet's \(K=0\) — this is the everyday fact that maps of the round Earth cannot be flat without distortion, and the reason orange peel tears when flattened.

Geodesic curvature

We may take the curve to pass through the origin of an orthogonal system; the curve projected onto the tangent plane, called \(\vect{x}_{g}\), is then given by

where \(\hat{T}'\) and \(\hat{N}'\) span the tangent plane at the point of tangency. Differentiating with respect to the arc-length parameter of the curve on the surface,

On the other hand, from Equation (13.39), the normal curvature vector of the projected curve is given by

where \(\hat{T}_{g}\) is the unit tangent vector defined in Equation (13.38) and \(s_{g}\) is the arc-length parameter defined in Equation (13.35), both for the projection curve. By the chain rule we may write

From Equation (13.37) we see that

so that

Furthermore, from the definition of the unit tangent vector (Equation (13.38)),

Again by Equation (13.37), substituting in Equation (13.116) and differentiating,

Finally, since in this case we are considering that at the point of tangency \(\hat{T}=\hat{T}'\), from Equations (13.113) and (13.114),

and from Equations (13.40) and (13.42),

Hence Equation (13.117) reduces to

Considering Equation (13.97), we see that

Alternative expression for the geodesic curvature vector.

Consider a curve on a surface \(\vect{x}\) parametrised by arc length (see Equation (13.56)). From the definition of the normal curvature vector of the curve (Equation (13.39)),

Comparing with Equation (13.119), we see that

Geodesic lines

Consider two points on a surface, and a curve on the surface containing them. Among all curves satisfying this, there must be one of least arc length, as given by Equation (13.67). Intuitively, such a curve possesses a projection curve with vanishing geodesic curvature.

By the calculus of variations it can be proved that Equation (13.121) results from extremising the arc length given by Equation (13.67), as the following proposition shows.

Geodesic parallelism

Consider a curve \(\vect{x}=\vect{x}(u_1(t),u_2(t))\) on the surface \(\vect{x}=\vect{x}(u_1,u_2)\). Assign to the curve a vector field \(\vect{A}\) of the form

At each point of the curve, \(\vect{A}\) is written as a linear combination of the partial tangent vectors of the surface, and therefore lies in the tangent plane. From vector analysis, the vectors \(\vect{A}\) are parallel for every value of \(t\) if and only if

Analogously, we say that the vectors \(\vect{A}\) are geodesically parallel for every value of \(t\) if and only if

From Equation (13.126),

Therefore the field \(\vect{A}\) is geodesically parallel if and only if

whence the condition of geodesic parallelism is

Holonomy and the local Gauss–Bonnet theorem

Transport a vector geodesically parallel all the way around a closed curve on a surface and it comes back to the point it started from — but not, in general, to the vector it started as. The failure is the holonomy of the curve, and the theorem of this subsection identifies it: it is the total Gaussian curvature enclosed. That is the local form of the Gauss–Bonnet theorem, and it is what makes curvature an observable rather than a bookkeeping device, since the angle it predicts can be measured by carrying a gyroscope, or a swinging pendulum, around a circuit.

Differentiable manifolds

The definitions given below serve to introduce the concept of a differentiable manifold, which results from generalising the idea of curves and surfaces. The definition is built up from the concept of a topological space (Topological and Metric Spaces), which we progressively equip with the pertinent structure so as to obtain an object rich in properties that can be linked with nature. One of the key concepts is that of a coordinate system, which is in intimate correspondence with reference frames: the tool of theoretical physics that lets us fix a point in the Universe with respect to which a physical system is being evaluated.

Charts, atlases, and differentiable structure

Coordinate systems

Consider a topological space. In the context of this chapter we refer to the elements of a topological space as points.

Let \(M\subset\R^{N}\) be a topological space and \(U\subset\R^{n}\) with \(n\leq N\). Consider a point \(P\in M\) and \(x=(x^{1},\ldots,x^{n})\in U\), and consider further the injective map \(\phi: x\longmapsto\phi(x)=P\).

Intuitively, \(\phi\) must respect the topology of \(M\) in order to describe its points correctly through coordinates. To this end we must demand that points of \(M\) be close if and only if their coordinates are. The idea is to require

where \(x_{l}\) is a sequence of points in \(U\) and \(P_{l}=\phi(x_{l})\) a sequence of points in \(M\). Demanding “\(\Leftarrow\)” we meet the continuity of \(\phi\), and demanding “\(\Rightarrow\)” we meet the continuity of \(\phi^{-1}\). We further wish a one-to-one relation between the points of \(U\) and (at least some of, since in general more than one set like \(U\) is needed to describe all the points of \(M\)) the points of \(M\), which suggests that \(\phi\) must be bijective. In conclusion, the topology is respected if and only if \(\phi\) is a homeomorphism. By the definition of continuity, demanding “\(\Leftarrow\)” the set \(\phi^{-1}(\phi(U))=U\) must be open, and demanding “\(\Rightarrow\)” the set \((\phi^{-1})^{-1}(U)=\phi(U)\) must be open. The geometric interpretation is that (by the definition of an open set) we can place an open ball centred at \(P\) and one centred at \(x\),

from which we see that the points neighbouring \(x\) should give us information about the points neighbouring \(P\) (by the definition of a neighbourhood), which happens if and only if the map \(\phi\) is smooth.

Coordinates

As we have seen, given a coordinate system there is a one-to-one relation between the points of \(U\) and some points of \(M\). The variables \((x^{1},\ldots,x^{n})\) corresponding to a point \(P\in M\) receive the name of coordinates of the point \(P\) with respect to the coordinate system \((\phi,U)\). To symbolise (and not forget) the exact point of \(M\) that we wish to describe by means of \((\phi,U)\), we denote the coordinates of a point \(P\in M\) as \((x^{1},\ldots,x^{n})\stackrel{\text{not.}}{=}x^{i}(P)\) with \(i=1,\ldots,n\). Thus, when we want to refer to the point \(x\in U\) we write \(x\) or \((x^{1},\ldots,x^{n})\); when we want to refer explicitly to the point \(P\) through its coordinates, we write \(x^{i}(P)\); and for the coordinates of arbitrary points of \(M\) (within the codomain of \(\phi\)) we use \(x^{i}\). In summary,

Even though we have insisted on the difference between the quantities in \(U\) and the coordinates of the point, numerically they are the same, which is useful in practice; in the theory, however, it is necessary to be rigorous with the notation. It is very important to note that the coordinates of a point depend on the subset \(U\) of \(\R^{n}\) we have chosen. Note that \(\phi: U\longrightarrow\phi(U)\) is bijective, so this one-to-one relation characterises the points of \(\phi(U)\) by their coordinates in \(\R^{n}\). Note also that a correct notation for the point \(P\) in terms of its coordinates in \(U\) would be \(P=\phi(x)=\phi(x^{1},\ldots,x^{n})\), but not \(P=\phi(x^{i})\) nor \(P=\phi(x^{1}(P),\ldots,x^{n}(P))\).

Atlases

Consider a family of \(n\)-dimensional coordinate systems \(\set{(\phi_{a},U_{a})}_{a\in I}\) that cover the whole space \(M\), that is,

such that \(U_{a}\cap U_{b}\neq\varnothing\) for \(a\neq b\).

Let the overlap regions be

The sets \(\phi^{-1}_{a}(\Omega_{ab})\) and \(\phi^{-1}_{b}(\Omega_{ab})\) are open because \(\phi_{a}\) and \(\phi_{b}\) are homeomorphisms. We further define the transition maps as

The transition maps are continuous, being compositions of continuous maps, and since the sets \(\phi^{-1}_{a}(\Omega_{ab})\) and \(\phi^{-1}_{b}(\Omega_{ab})\) are open, the transition maps are moreover homeomorphisms. A point \(P\) may be described by \(P=\phi_{a}(x^{1}_{a},\ldots,x^{n}_{a})\) or by \(P=\phi_{b}(x^{1}_{b},\ldots,x^{n}_{b})\). This means we can perform a change of coordinate system of the form

which respects the topology of \(M\) because the \(\phi_{ba}\) are homeomorphisms. For the information about the points neighbouring \(P\) to be well defined after changing the coordinate system, we must demand that the transition maps be smooth.

Differentiable maps on a topological space

Consider now a topological space \(M\) equipped with an atlas \(\set{(\phi_{a},U_{a})}_{a\in I}\), and let

be a map. A key question is: under what conditions is \(f\) differentiable? Since we know the usual definition of differentiability at a point \(x\in\R^{n}\), we wish to extend it to the points of a topological space by virtue of the given atlas. Of course, we know that a map that is not continuous is not differentiable, so at the very least we must demand that \(f\) be continuous. Choose the coordinate system \((\phi_{a},U_{a})\) and define the map \(\bar{f}_{a}=f\circ\phi_{a}\).

The map \(\bar{f}_{a}\) is continuous, being a composition of continuous maps; moreover, the atlas allows us to define \(f\) on a neighbourhood of \(P\), so that \(\phi_{a}\) is well defined on a neighbourhood of \((x^{1}_{a},\ldots,x^{n}_{a})\), since \(\phi_{a}\) respects the topology (again a consequence of working with an atlas). Thus our map \(\bar{f}_{a}\) is in a position to be differentiable.

But what would happen if we changed to a coordinate system \((\phi_{b},U_{b})\) and \(\bar{f}_{b}=f\circ\phi_{b}\) were not differentiable at \(P\)? This does not occur, since

and thus, since \(\phi_{ab}\) is smooth, it is at least of class \(C^{1}\) and hence differentiable; and since \(\bar{f}_{a}\) is differentiable, \(\bar{f}_{b}\) is differentiable, being a composition of differentiable maps. The atlas has again entered the game: it is the smoothness condition in the atlas that allows us to define a differentiable map on \(M\).

Differentiable structure

Consider a topological space now equipped with the atlases \(\set{(\phi_{a},U_{a})}_{a\in I}\) and \(\set{(\psi_{b},V_{b})}_{b\in J}\). The relation

is an equivalence relation.

Note the following: all the coordinate systems of two compatible atlases are smoothly related, because their union is an atlas (and the coordinate systems of an atlas are smoothly related).

Differentiable manifold

It is important to discuss the following. The dimension of the manifold refers to the number of coordinates we need to describe completely an arbitrary point of it. The definitions have been constructed in such a way that it is not possible to have an \(n\)-dimensional manifold in which some points are described by fewer than \(n\) coordinates.

Diffeomorphism

The notion of a differentiable map developed above compares a manifold with \(\R\); nothing prevents us from comparing two manifolds with each other.

That the condition need be checked in one pair of charts only is the computation already performed for \(\bar{f}_{a}\) and \(\bar{f}_{b}\) above: changing either chart composes Equation (13.140) with a transition map, which is smooth because we are working inside atlases. Definition 13.45 is the special case \(N_{n}=E_{1}\) read in the Cartesian chart.

The differential of a smooth map

Immersion, embedding, and submanifold

The image of an embedding is an embedded submanifold: transporting the differentiable structure of \(M_{m}\) through the homeomorphism \(f: M_{m}\longrightarrow f(M_{m})\) produces an atlas on \(f(M_{m})\) whose charts are the \(f\circ\phi_{a}\), and with respect to it \(f\) becomes a diffeomorphism onto its image. The third condition in Definition 13.53 is not implied by the first two, as the following example shows.

Product of manifolds

Hypersurfaces

Hypersurfaces are produced in practice by one scalar equation, and the theorem that licenses this is the following.

Full derivation in Appendix A.

Derives Equation (13.146).

Higher codimension and maps of constant rank

Theorem 13.59 cuts out a submanifold with one equation. Physics almost never presents one equation: a constraint surface in phase space, the level set of a momentum map, the shell of a conserved four-vector are all defined by several. The extension is not automatic — “at least one partial derivative is nonzero” has to become a rank condition — and the general statement is the following.

The rank of \(F_{*}\) need not be maximal, and when it is merely constant the map still has a normal form. This is the statement that turns “the image of a degenerate Legendre map” or “the set swept out by a family of solutions” into a submanifold with a definite dimension, and it is proved from the same implicit function theorem.

Group actions and quotient manifolds

The other standard way a manifold arises is not by cutting a bigger one with equations but by identifying its points: the configuration space of a rigid body is the set of frames modulo nothing, but the space of shapes is a set of configurations modulo rotations, and a reduced phase space is a level set modulo a symmetry group. The question is when the set of orbits is again a manifold, and the answer needs two hypotheses that are easy to state and easy to forget.

Full derivation in Appendix A.

Derives Theorem 13.67.

Euclidean space

The Cartesian coordinate system

Consider the topological space \(M=\R^{n}\). Note that the pair \((\id_{\R^{n}},\R^{n})\) defines a coordinate system, for \(\id_{\R^{n}}\) is trivially a smooth homeomorphism: it is the bijective map that associates with each point \(x\in\R^{n}\) the point \(x\in\R^{n}\) itself.

We say that \((\id_{\R^{n}},\R^{n})\) is a Cartesian coordinate system of \(\R^{n}\). In this coordinate system the coordinates of a point \(P\in\R^{n}\) are \(x^{i}(P)\), which are called the Cartesian coordinates of the point \(P\) (with respect to the coordinate system \((\id_{\R^{n}},\R^{n})\)).

Euclidean space $E_{n}$

In the preceding context, a single Cartesian coordinate system suffices to cover all of \(\R^{n}\), so \(\set{(\id_{\R^{n}},\R^{n})}\) defines an \(n\)-dimensional atlas and, trivially, a differentiable structure. We conclude that \(\R^{n}\) is an \(n\)-dimensional differentiable manifold, which we denote \(E_{n}\) and call Euclidean space (or the Euclidean manifold); we also say that spaces of this type possess a Euclidean geometry. The fact that a single coordinate system describes the whole Euclidean space leads us to consider describing a manifold, such as a curve or a two-dimensional surface, as a submanifold of \(E_{n}\).

We say that the origin of \(E_{n}\) is the point \(O=\id_{\R^{n}}(0,\ldots,0)\) (with respect to the coordinate system \((\id_{\R^{n}},\R^{n})\)); therefore \(x^{i}(O)=0\) for all \(i\).

Pictorial representation

We represent pictorially a Cartesian coordinate system and a point \(P=\id_{\R^{n}}(x)\in E_{n}\) of coordinates \(x^{i}(P)\) as a diagram of perpendicular axes[That the axes are perpendicular has a mathematical background studied further below. This representation is due to René Descartes.]. For the case \(n=2\) it has been historically common to denote the coordinates of a generic point, \(x^{i}\), as \((x,y)\):

where \(E_{2}\) is called flat two-dimensional space. For the case \(n=3\) it has been historically common to denote the coordinates of a generic point, \(x^{i}\), as \((x,y,z)\):

where \(E_{3}\) receives the name flat three-dimensional space.

For the case \(n=2\), the Euclidean space \(E_{2}\) works very well for describing plane geometric figures, for example a square: given a square in the plane, we can specify each of its points by its coordinates. The same occurs with solid geometric figures in the space \(E_{3}\).

Vectors in $E_{n}$

To refer to the position vector of arbitrary points of \(E_{n}\) we make no reference to the point, denoting it simply \(\vect{x}\). The position vector of the origin is denoted \(\vect{0}\).

Cartesian coordinates as components of the position vector

The fact that a single coordinate system describes the whole Euclidean space provides us with the bijective maps \(\vect{x}: P\longmapsto\vect{x}_{P}\) and \(x^{i}: P\longmapsto x^{i}(P)\); that is, the map assigning to each point of \(E_{n}\) its position vector (with respect to \(O\)), and the map assigning to each point of \(E_{n}\) its coordinates. A point of \(E_{n}\) is determined by its position vector or by its Cartesian coordinates, so we may establish a relation between the two concepts.

We say that the Cartesian components of the position vector of a point \(P\) with respect to \(O\) are the Cartesian coordinates of \(P\) (with respect to the coordinate system \((\id_{\R^{n}},\R^{n})\)), which we symbolise as

(for a generic point we write \(\left(\vect{x}\right)^{i}=x^{i}\)). Note that this relation is one-to-one: given a coordinate system and an origin, the components of the position vector of a point are unique. With respect to the maps just mentioned we have, moreover, \(O\longmapsto\vect{0}\) and \(O\longmapsto(0,\ldots,0)\) respectively. In this sense, giving a coordinate system amounts to giving an origin (and vice versa) with respect to which we can locate points in \(E_{n}\).

$E_{n}$ as a vector space

As we know, the set of all points of \(\R^{n}\) forms an \(n\)-dimensional vector space. Moreover, each of these points gives the Cartesian components of the position vector of the corresponding point of Euclidean space \(E_{n}\). We may therefore picture the vector space \(\R^{n}\) as follows:

Since the relation between the position vector of \(P\) (with respect to a given origin) and the coordinates of \(P\) (with respect to a given coordinate system) is one-to-one, from now on, given a Cartesian coordinate system (and hence an origin), we refer to the points of \(E_{n}\) by means of position vectors. This is only a notation to make the equations more transparent. Since we can now conceive the manifold \(E_{n}\) as a vector space — that is, position vectors as vectors of a vector space — we can define the Euclidean (or standard) inner product on \(E_{n}\) as

which clearly satisfies the axioms of an inner product. We further define the Euclidean (or standard) norm of \(E_{n}\) as the norm associated with the Euclidean inner product,

and, lastly, the Euclidean (or standard) metric of \(E_{n}\) as the metric associated with the Euclidean norm,

Displacements

Consider two points \(P\) and \(Q\) in Euclidean space, with position vectors \(\vect{x}_{P}\) and \(\vect{x}_{Q}\) with respect to a given origin. In this context it is common to symbolise the vector \(\overrightarrow{PQ}\) as

or, for two arbitrary points, simply as \(\delta\vect{x}\), which is called the displacement vector. It serves to denote the change of position between two points.

In the preceding context, one then analyses the case in which the Euclidean distance between these points is infinitesimally small.

Curves in Euclidean space

Differentiable curves

We therefore need only one coordinate to describe a curve locally — we say locally because in general more than one coordinate system is needed to describe it globally, though each of them involves a single coordinate. Thus the possible coordinate systems needed to describe the curve are of the form

The variable \(\lambda\) is a coordinate associated with the coordinate system \((\gamma,\left]a,b\right[\,)\), and \(\gamma(\lambda_{P})\), with \(\lambda_{P}\in\left]a,b\right[\), is a point of the curve. For a generic point of the curve we may identify \(\gamma(\lambda)\) with its position vector \(\vect{x}_{\gamma(\lambda)}\). The components of the position vector are therefore the quantities

which are the (Cartesian, since the curve lives in \(E_{n}\)) coordinates of the point \(\gamma(\lambda)\in\mathcal{C}\subset E_{n}\). The coordinate \(\lambda\) parametrises the curve: for each value of \(\lambda\in\R\) we have a position vector \(\vect{x}=\vect{x}(\lambda)\) characterising a point of the curve.

It is necessary to clarify the difference between the two statements: \(\lambda\) is the coordinate of the point \(\gamma(\lambda)\in\mathcal{C}\), while the \(x^{i}(\gamma(\lambda))\) are the Cartesian coordinates of the point \(\gamma(\lambda)\in E_{n}\). Now, as mentioned before, in general it is not possible to describe a curve completely with a single coordinate system (we need an atlas). The problem of needing more than one coordinate system is not transcendental for calculations (though it does bring topological consequences to be studied in theoretical physics). Let us illustrate with a well-known and much-cited example: the circle of radius \(R\) as a hypersurface of \(E_{2}\), \(S_{1}=\set{(x,y)\in\R^{2}\mid x^{2}+y^{2}=R^{2}}\). We wish to describe by coordinate systems all points of \(E_{2}\) satisfying \(x^{2}+y^{2}=R^{2}\). A possible candidate is \((\gamma,\left]0,2\pi\right[\,)\), where

with the Cartesian coordinates of \(\gamma(\phi)\) given by

This coordinate system describes almost all of the desired region — almost, because the point \((R,0)\) is not included.

If we put the interval \(\left[0,2\pi\right[\) or \(\left]0,2\pi\right]\) in the domain of \(\gamma\) we would include the point, but in the definition of a differentiable manifold we worked only with open domains, to make sure we respect the topology (the maps of a coordinate system must be homeomorphisms). We could contemplate an interval \(\left]0,2\pi+\delta\right[\) so as to include the mentioned point, but then the map \(\gamma\) would no longer be bijective, which as we know is a crucial property of \(\gamma\) as a homeomorphism, meant to preserve the topology. How can we fix this problem? Is there a coordinate system that describes all the points of the circle? The answer is no, and as we shall see this is related to the fact that the circle is a compact subset of \(E_{2}\). If a single coordinate system \((\gamma,I)\) described every point of the circle, the homeomorphism \(\gamma: I\longrightarrow S_{1}\) would force \(I\) to be compact

— a contradiction, since \(I\) must necessarily be open.

We define the coordinate systems \((\gamma_{1},\left]-\delta,\pi+\delta\right[\,)\) and \((\gamma_{2},\left]\pi-\delta,2\pi+\delta\right[\,)\) with

where \(x^{i}(\gamma_{1}(\phi))=(R\cos\phi,R\sin\phi)\) with \(\phi\) varying in \(\left]-\delta,\pi+\delta\right[\), and \(x^{i}(\gamma_{2}(\phi))=(R\cos\phi,R\sin\phi)\) with \(\phi\) varying in \(\left]\pi-\delta,2\pi+\delta\right[\). We mention that we have used the same variable \(\phi\) for both coordinate systems because it is a dummy variable and there is no ambiguity (it merely happens that in both coordinate systems the curve behaves locally in the same way). The two coordinate systems cover the whole circle.

Thus \(\set{(\gamma_{1},\left]-\delta,\pi+\delta\right[\,), (\gamma_{2},\left]\pi-\delta,2\pi+\delta\right[\,)}\) defines an atlas on \(S_{1}\), which clearly possesses a one-dimensional differentiable structure. This means that \(S_{1}\) is a hypersurface of \(E_{2}\).

An example of a non-compact curve that can be described by a single coordinate system is the straight-line segment between two points \(P,Q\in E_{n}\), denoted \(L_{PQ}\). The straight line joining the two points is completely described by the coordinate system \((\gamma,\left]0,1\right[\,)\) with

where \(x^{i}(\gamma(\lambda))=(1-\lambda)x^{i}(P)+\lambda x^{i}(Q)\) with \(\lambda\) varying in \(\left]0,1\right[\). Visually, the curve “grows” from \(P\) to \(Q\) as \(\lambda\) grows in \(\left]0,1\right[\). Note further that \(x^{i}(\gamma(\lambda\to 0))=x^{i}(P)\) and \(x^{i}(\gamma(\lambda\to 1))=x^{i}(Q)\).

Admissible change of parameter

Given a curve \(\mathcal{C}\), described by \(\vect{x}=\vect{x}(\lambda)\), we can perform coordinate changes in our coordinate systems of the form seen in Equation (13.163). A general coordinate transformation may take the form \(\bar{\lambda}=\bar{\lambda}(\lambda)\), and, as we know, the map

must be a smooth function of \(\lambda\) to guarantee that we possess a differentiable structure. The new coordinate system is then of the form \((\bar{\gamma},\left]c,d\right[\,)\) where

with \(\bar{\gamma}=\gamma\circ\bar{\lambda}\).

It is important to point out that an admissible change of parameter (a general coordinate transformation of the single coordinate describing the curve locally) must not be confused with a general coordinate transformation of the Euclidean space in which the curve is embedded. There is, however, no problem in performing a general coordinate transformation of the Euclidean space in order to describe the curve. As an example of an admissible change of parameter, let the straight-line curve \(L_{PQ}\) with \(P,Q\in E_{2}\) be described by

with \(\lambda\) varying in \(\left]0,1\right[\). We may perform the change of parameter \(\bar{\lambda}(\lambda)=1-\lambda\), which is clearly admissible, with \(\bar{\lambda}\) still varying in \(\left]0,1\right[\). The straight line is now described by

Visually, this change of parameter makes the curve “grow” from \(Q\) to \(P\) instead.

Tangent vector

Consider a differentiable curve in \(E_{n}\) described locally by \(x^{i}=x^{i}(\lambda)\), \(i=1,\ldots,n\), with respect to \((\gamma,\left]a,b\right[\,)\), or equivalently with respect to the basis \(\set{\hat{x}_{i}}\) of \(E_{n}\) as a vector space. We define the tangent vector to the curve at the point \(P=\gamma(\lambda_{P})\) as the vector

that is, as the variation of the coordinates of the curve as the parameter varies.

Arc element and length of a curve

Section 13.3.1 built the theory of curves extrinsically: the curve was a map into \(\R^{3}\), and its tangent, normal and binormal were vectors of the ambient Euclidean space, differentiated componentwise. The same content can be stated using nothing but the metric of the space the curve lives in and the covariant derivative it determines — the intrinsic formulation, which is the one that survives when the ambient space is removed. Throughout this stretch \((M,g)\) is a Riemannian manifold of dimension \(D\) (Definition 13.117 with \(q=0\)), with coordinates \(x^{\mu}\) and Levi-Civita connection \(\mathring{\Gamma}\) (Theorem 13.150); the reader who prefers a concrete setting may read \(M=E_{3}\) in arbitrary curvilinear coordinates, in which case \(g_{\mu\nu}\) is the Euclidean metric written in those coordinates.

The arc element is the line element Equation (13.259) of the displacement along the curve,

which for a surface in \(E_{3}\) is exactly the first fundamental form Equation (13.61), the metric \(g_{ij}\) there being the induced metric Equation (13.261).

Curvature vector, binormal, and torsion

In \(E_{3}\) referred to Cartesian coordinates the Christoffel symbols vanish and Equation (13.171) collapses to \(\dd^{2}\vect{x}/\dd s^{2}\), which is the extrinsic curvature vector Equation (13.39); comparison with Equations (13.121) and (13.297) shows that the curves with \(k^{\mu}=0\) are precisely the geodesics.

Where \(\kappa>0\) we may therefore define the unit principal normal \(N^{\mu}=k^{\mu}/\kappa\), orthogonal to \(u^{\mu}\), in parallel with Definition 13.14. For \(D=3\) — the case treated extrinsically in Section 13.3.1, and the observed spatial dimension — the orthogonal complement of \(\gen{\set{u,N}}\) in each tangent space is one-dimensional, and we fix the unit binormal \(B^{\mu}\) as the unique unit vector completing \((u,N,B)\) to a positively oriented orthonormal frame; equivalently, with the Levi-Civita tensor Equation (13.257),

which reduces to the cross product Equation (13.44) in Cartesian coordinates.

Tensor analysis on manifolds

Vectors, covectors, and tensors on a manifold

The definition of a vector on a manifold

The definition of a vector in \(\R^{N}\) is an oriented straight-line segment joining two points. This tells us that a vector as we know it is a bilocal object. Would this definition remain consistent if the space no longer had Euclidean geometry — that is, on a general manifold? Consider an \(n\)-dimensional manifold \(M\subset\R^{N}\). The Euclidean metric of \(\R^{N}\) allows us to define straight lines joining points on the manifold, but such straight lines are clearly not contained in \(M\).

Along which route, then, should the line between \(A\) and \(B\) be traced? There is no possible route unless we define beforehand a metric on the space \(M_{n}\), for without this concept we cannot have a concept of straight line. For this reason we cannot define a vector on the manifold as a bilocal object: we must develop a formalism to understand a vector as a local object.

Let \((\gamma,\left]a,b\right[\,)\) be a coordinate system describing a curve contained in \(M\). The curve is characterised by the position vectors \(\vect{x}_{\gamma(\lambda)}\) with \(\lambda\) varying in \(\left]a,b\right[\). Denote by \(\lambda_{0}\) the value of the parameter at which the point of the curve is \(P=\gamma(\lambda_{0})\in M\).

We say that the vector \(V_{P}\in\R^{N}\) is a tangent vector to \(M\) at \(P\) if and only if it is tangent to the curve at \(P\), that is, if its components with respect to the coordinate system \((\gamma,\left]a,b\right[\,)\) are

The tangent vector \(V_{P}\) must not be viewed as a displacement lying in \(M\), but as lying in some kind of tangent surface coinciding with \(M\) at \(P\). Since \(M\) is equipped with an atlas, we can define a differentiable map of the form \(f: M\longrightarrow\R\). Then \(f\circ x:\R\longrightarrow\R\) is differentiable, being a composition of differentiable maps, and

But which map \(f\) do we choose? It does not matter. What matters is to see how the curve generates changes in \(f\). Indeed, \(f\circ x:\R\longrightarrow\R\) allows us to evaluate the changes in \(f\); these changes are given by the directional derivative (Equation (13.178)) in the direction of the curve, that is, in the direction of the vector \(V_{P}\).

We may conclude that the operator generating the changes of \(f\) in the direction of \(V_{P}\) is \(\left.\dv{}{\lambda}\right|_{\lambda_{0}}\). By the chain rule and Equation (13.177),

Therefore \(\left.\dv{}{\lambda}\right|_{\lambda_{0}}\) can be written as a linear combination of the basis \(\set{\pp_{i}}\), with components exactly \(V^{i}_{P}\).

We must then symbolise a vector field without reference to the point of the manifold, since we are considering all of them. In the context of the definition of a vector, we symbolise a vector field \(V\) as

When no reference to the point is made, it is understood that we are speaking of a field.

Tangent space and coordinate basis

Owing to the linearity of the derivative, and since it has the properties of addition and multiplication by a scalar, the set of all vectors tangent to \(M\) at \(P\) forms an \(n\)-dimensional vector space, which we call the tangent space and denote \(T_{M}(P)\) (for an arbitrary point, \(T_{M}\)). This may seem inconceivable, but so it is: at each point of the manifold we can define a vector space.

As discussed in the definition of a vector on a manifold, we can write a vector \(V\) on \(M\) as a linear combination of the vectors \(\pp_{i}\),

This means that the set \(\set{\pp_{i}}^{n}_{i=1}\) forms a basis of the tangent space, which we call the coordinate (or holonomic) basis. We therefore have

The vectors of the coordinate basis are tangent to the coordinate lines.

For example, the coordinate-basis vectors of \(E_{2}\) in Cartesian coordinates are \(\pp_{x}\) and \(\pp_{y}\), while in polar coordinates they are \(\pp_{r}\) and \(\pp_{\phi}\). The relation between the Cartesian and polar coordinates describing \(E_{2}\) is

so that

and likewise

whence

Covectors and the cotangent space

From the theory of vector spaces we know that the set of all linear functionals defined on a vector space forms a vector space, called the dual. Thus the set of all covectors defined at a point \(P\in M\) forms the vector space

which we call the cotangent space at \(P\). As shown below (Equation (13.239)), the differentials \(\dd x^{i}\) furnish the basis dual to the coordinate basis, so that

Tensors

We say that a tensor is contravariant if it is of type \((r,0)\) with \(r\in\N\), and covariant if it is of type \((0,s)\) with \(s\in\N\). For example, a vector is a contravariant tensor of type \((1,0)\) and a covector is a covariant tensor of type \((0,1)\).

Pullback and pushforward

If we wished to compare two vectors assigned at distinct points of a differentiable manifold, we could not simply subtract them, since they belong to different vector spaces. We must provide a mechanism that maps one of the vectors into the space of the vector with which we wish to compare it.

Consider a diffeomorphism \(f: M\longrightarrow M\). For a point \(P\), the map \(f\) induces the pushforward \(f_{*}: T_{M}(P)\longrightarrow T_{M}(f(P))\). This amounts to taking a tangent vector at \(P\) and locating it at \(f(P)\).

One also has the pullback \(f^{*}: T^{*}_{M}(f(P))\longrightarrow T^{*}_{M}(P)\), which to each covector at \(f(P)\), say \(\omega_{f(P)}\), assigns by definition the covector \(f^{*}(\omega_{f(P)})=\omega_{f(P)}\circ f_{*}\) at \(P\). Covectors are linear functionals acting on vectors of the tangent space: for a tangent vector \(V_{P}\) at \(P\),

Vector fields on basic manifolds

Tensors under general coordinate transformations

General coordinate transformations

We call a coordinate system \(K\) a set of functions \(\set{x^{i}}^{n}_{i=1}\) such that the location of a point \(P\) is given by \(x^{i}(P)\).

For practical purposes we assume that these \(n\) functions of \(n\) variables are of class \(C^{2}\).

In the context of coordinate systems it is necessary to extend the definition of the Kronecker delta to every coordinate system: we define the Kronecker delta symbol as the quantities

in every coordinate system.

A GCT is linear if and only if it can be written as \(\bar{x}^{i}=\Lambda^{i}{}_{j}x^{j}\) with a constant matrix \(\Lambda^{i}{}_{j}\) (compare Definition 13.2).

Transformation law of a vector

Since a vector field is the coordinate-independent object \(V = V^{i}\pp_{i}\), its components in two coordinate systems obey

where both sides have been applied to the coordinate function \(\bar{x}^{j}\); hence

Contravariant, covariant, and mixed tensors

We call contravariant vector a tensor of type \((1,0)\); from Equation (13.200), its components with respect to another coordinate system are given by

We call covariant vector a tensor of type \((0,1)\); from Equation (13.201),

The displacement vector

For a GCT one cannot say that the coordinates themselves transform as a vector would: under a GCT the coordinates are neither contravariant nor covariant. Differentiating the relations Equation (13.197), however, we see that

so the differential of the coordinates transforms as a contravariant vector, which we call the displacement vector.

Scalars and invariance

The notion of the rank of a tensor under general transformations is the number of indices it carries, and the number of times the Jacobian matrix of the transformation must be applied to carry out a change of coordinates. If we apply this matrix zero times, we obtain the same quantities in both coordinate systems, since we are not transforming them at all. We may extend the definition to tensors of rank \(0\) by defining these invariant quantities: quantities that depend on position with respect to a coordinate system but take the same value in every coordinate system. That is,

More generally, we call invariant tensor a tensor satisfying

We call scalar an invariant tensor of rank \(0\).

Scalar product

Consider the components of a contravariant vector \(T^{i}\) and of a covariant vector \(U_{l}\). With respect to \(\bar{K}\),

Consider the quantity

from which we see that this type of product is a scalar under GCTs. We call this multiplication of vectors the scalar product.

The Kronecker delta as a mixed tensor

From its definition, the Kronecker delta is an invariant tensor. Observe that

Thus, from Equation (13.202), the Kronecker delta satisfies the definition of a tensor of type \((1,1)\).

Symmetry and antisymmetry

We say that a tensor with components \(T^{i_1\cdots i_r}{}_{l_1\cdots l_s}\) is a tensor symmetric with respect to the indices \(i_{p}\) and \(i_{q}\) if and only if

or a tensor symmetric with respect to the indices \(l_{p}\) and \(l_{q}\) if and only if

The quantities so defined are tensors under GCTs, by the permutation of indices. Analogously, the tensor is antisymmetric with respect to a pair of upper indices, or a pair of lower indices, if the interchange of the pair produces a minus sign; again the quantities so defined are tensors under GCTs.

Consider now a tensor of type \((1,1)\), and suppose that in some coordinate system

that is, quantities symmetric with respect to one contravariant and one covariant index. Then

from which we see that this type of symmetry does not generate tensors under GCTs. Symmetry or antisymmetry properties must therefore always be taken with respect to two contravariant or two covariant indices; for this reason we display them below only for covariant indices.

The symmetry or antisymmetry of a rank-\(2\) tensor reads

which in matrix notation is equivalent to

Symmetric and antisymmetric parts

Adding Equations (13.213) and (13.214) we see that

so a rank-\(r\) tensor can be decomposed into its symmetric and its antisymmetric part. For a tensor of order \(2\),

The vector-product tensor

We define the vector-product tensor as

From the algebra of tensors, the quantities \(C_{ij}\) are the components of a tensor of type \((0,2)\) under GCTs. Moreover, from Equation (13.217),

so that

that is, the vector-product tensor is an antisymmetric tensor of order \(2\).

Gradient of a scalar

Consider Equation (13.206) and apply the partial derivative with respect to the coordinate \(\bar{x}^{i}\):

from which we see that

that is, the gradient of a scalar is a covariant vector under GCTs.

A Jacobian identity

From the chain rule,

where \(\abs{\pp\bar{x}/\pp x}\) denotes the Jacobian determinant of the transformation.

Tensor densities

The quantities met so far transform under a GCT by factors of \(\pp\bar{x}/\pp x\) alone. A second family of objects, indispensable as soon as one integrates over a manifold, transforms by those factors and by a power of the Jacobian determinant.

The sources develop none of this in general and treat only the following three-dimensional instances.

The pseudo-vector cross product

Consider the vector-product tensor (Equation (13.217)). Since this tensor is totally antisymmetric, we can define the associated dual pseudo-tensor, in three dimensions, as

which we call the pseudo-vector cross product, or cross product of two vectors. From Equation (13.217),

so that

The source calls Equation (13.225) a pseudo-vector and stops there. The name records a property, and the property has to be established; the three results below do that, and supply at the same time the elementary identities of the three-dimensional cross product, which the sources use without stating and which the physical parts of this treatise cref constantly.

Stokes' theorem for an antisymmetric tensor field

For an antisymmetric tensor field \(H_{ij}\) with associated vector \(H_{i}\),

where \(\dd f^{ij}\) is the antisymmetric surface element of \(S\) and \(\pp S\) its boundary. The source states the relation without derivation; it is the flat three-dimensional instance of the general Stokes theorem for differential forms, and both are proved in the appendix.

Full derivation in Appendix A.

Derives Equation (13.231).

Differential forms: $k$-forms and $k$-vectors

$k$-forms and $k$-vectors

The set of all \(k\)-forms at \(P\) forms a vector space, which we denote \(\Lambda^{k}_{M_n}(P)\). Of course,

Analogously, the set of all \(k\)-vectors forms a vector space, denoted \({}^{*}\Lambda^{k}_{M_n}(P)\), with

Both assertions, and the dimension of the two spaces, follow from a single count.

The basis of the cotangent space

Let \(P\in M_{n}\), with \(M_{n}\) a differentiable manifold, and let \(f: M_{n}\longrightarrow\R\) be a differentiable map. Consider the map

We claim that \(\dd f(V_{P}) = V_{P}(f)\). Indeed, \(\dd f\) is defined as

and, recalling the discussion of the definition of a vector, \(\set{\pp_{i}}\) is a basis of the tangent space at \(P\). Thus

From Equation (13.238), taking the vectors \(\pp_{i}\in T_{M_n}(P)\) and \(f = x^{j}\), we see that

so \(\set{\dd x^{i}}\) is the basis dual to \(\set{\pp_{i}}\), that is,

and in particular the \(\dd x^{i}\) are covectors, that is, \(1\)-forms. From this we see that for the set of all tensors of type \((0,k)\),

The wedge product

Wedge product and change of basis

Observe the following. The space of all \(2\)-forms, \(\Lambda^{2}_{M_n}(P)\), can be generated by the basis \(\set{\dd x^{i}\otimes\dd x^{j}}\), but we must be careful: the direct product of two \(2\)-forms is not necessarily a \(4\)-form, since the result is a tensor of type \((0,4)\) that is not necessarily antisymmetric; that is, it can happen that

For closure — referring to the antisymmetry property — we must define a new type of product. As we now show, this can be achieved by a change of basis. In the basis already discussed for the cotangent space,

where the antisymmetry of \(\psi_{ij}\) was used in the second step.

Therefore

In short, we have performed the change of basis

Antisymmetry

From the definition of the wedge product it is immediate that

and clearly, taking \(j=i\) (no sum),

Powers of a $k$-form

We introduce the notation

which we call the \(m\)-th power of a \(k\)-form, with respect to the wedge product.

Commutation rule

Having defined the wedge product explicitly, we can analyse its commutativity. Let \(\psi\) be a \(k\)-form and \(\varphi\) an \(l\)-form. In the basis discussed above,

We thus obtain the commutation rule

Various properties follow from this relation by simple arithmetic. For example, consider the case \(\varphi=\psi\) with \(k\) odd. Since the square of an odd number is odd,

hence

The exterior derivative

It is clear that the exterior derivative is a linear operator, owing to the linearity of the ordinary derivative \(\pp_{j}\).

Nilpotency

Let \(\psi\) be a \(k\)-form. Then

Now, if \(\psi_{i_1\ldots i_k}\) is of sufficiently high differentiability class, \(\pp_{i}\pp_{j}\) is symmetric in \(i\) and \(j\), while \(\dd x^{i}\wedge\dd x^{j}\) is antisymmetric in those indices; hence their contraction vanishes. Thus

Leibniz rule

Let \(\psi\) be a \(k\)-form and \(\varphi\) an \(l\)-form. Then

where in the last step the \(1\)-form \(\dd x^{j}\) was carried through the \(k\) factors \(\dd x^{i_1}\wedge\cdots\wedge\dd x^{i_k}\) at the cost of a factor \((-1)^{k}\), by Equation (13.241). We conclude the identity

known as the Leibniz rule for differential forms.

Closed and exact forms

The converse is not automatic, and its validity depends on the topology of the domain. Two statements settle the matter for everything this treatise needs: on a star-shaped domain the converse holds outright, and in general it fails, with a completely explicit counterexample.

The symmetric-tensor analogue of Theorem 13.109 is the result that decides which strain fields of a continuum are realisable by an actual displacement, and it follows from the antisymmetric one by applying it twice.

The volume form

\(\eta\) is in fact an \(n\)-form, since

where \(\delta^{1\ldots n}_{i_1\ldots i_n}\) is the generalised Kronecker delta and \(\varepsilon_{i_1\ldots i_n}\) the Levi-Civita symbol. Hence \(\eta\) is an \(n\)-form with components

The Hodge dual

Consider a differentiable manifold \(M_{n}\). Recall that at each point \(P\) of the manifold there exist a tangent space \(T_{M_n}(P)\) and a cotangent space \(T^{*}_{M_n}(P)\), which are \(n\)-dimensional vector spaces, and that the set of all \(k\)-forms is a vector space \(\Lambda^{k}_{M_n}(P)\) with (Proposition 13.100)

and likewise for the set of all \(k\)-vectors, \({}^{*}\Lambda^{k}_{M_n}(P)\). Owing to the fact that

we can associate four \(\binom{n}{k}\)-dimensional vector spaces with each point of the manifold: the space of \(k\)-forms, the space of \((n-k)\)-forms, the space of \(k\)-vectors, and the space of \((n-k)\)-vectors. Recalling that two vector spaces of the same dimension are isomorphic, the four spaces mentioned are isomorphic; that is, we can guarantee that isomorphisms exist among them.

Which are the isomorphisms that move us between these spaces? The answer to this question is not in itself essential: we define the maps and prove their quality as isomorphisms when necessary.

We see that the change of components is given by

The metric tensor

A manifold, as defined so far, knows nearness but not distance: the charts of Section 13.4.1 let us differentiate, but nothing yet measures lengths, areas, volumes, or angles. As the source manuscript asks: the ambient \(\R^{N}\) of an embedded manifold carries the Euclidean metric, but can that law serve an arbitrary manifold? It cannot — a vector on a manifold lives in the tangent space at its point (Section 13.5.1), and each point carries its own. The remedy is a field that equips every tangent space with its own scalar product.

The metric and its inverse convert between the tangent and cotangent spaces — they lower and raise indices, \(V_{\mu} = g_{\mu\nu}V^{\nu}\), \(\omega^{\mu} = g^{\mu\nu}\omega_{\nu}\) — and define the invariant volume element

whose invariance follows from the Jacobian transformation of \(\det g\) (Section 13.5.3: \(\sqrt{\abs{g}}\) is a scalar density of weight one).

Isometries

The infinitesimal version of Equation (13.262) — the Killing equation — is derived in Section 13.9.2, and the manifolds whose isometry groups are as large as possible are classified in Section 13.13: they are precisely the flat, anti-de Sitter, and de Sitter geometries of Table 13.1.

Vielbein and local frames

The coordinate basis \(\pp_{\mu}\) diagonalizes differentiation, not the metric. For the physics of Part V — General Relativity and Cosmology — and for spinors, which transform under no representation of the diffeomorphism group — one needs at each point a basis in which the metric is the flat \(\eta_{ab}\) of Equation (13.1). Indices \(a,b,\ldots\) follow the conventions of Section 13.1.

Geometry in the frame formulation therefore has two covariances, acting on different indices per Section 13.1: diffeomorphisms on \(\mu,\nu,\ldots\) and local Lorentz transformations on \(a,b,\ldots\) — the structure that makes gravity resemble a gauge theory (Section 13.12).

Unlike the coordinate basis, the frame \(e_{a}\) is in general anholonomic: its Lie brackets (Section 13.9) do not vanish,

and the \(C^{c}{}_{ab}\) measure the failure of the frame to come from coordinates.

Flows, the Lie derivative, and Killing vectors

A vector field \(\vect{\xi} = \xi^{\mu}\pp_{\mu}\) generates a flow: the one-parameter family of diffeomorphisms \(\phi_{t}\) obtained by sliding every point along the integral curves \(\dv{x^{\mu}}{t} = \xi^{\mu}(x)\). The Lie derivative measures the rate of change of a tensor field as it is dragged by this flow, comparing the dragged tensor with the actual one:

It requires no metric and no connection — only the differentiable structure.

That opening sentence asserts something, and the assertion is a theorem: that a smooth vector field has a flow, that the flow is unique, that it is smooth in the point as well as in the parameter, and that each \(\phi_{t}\) is a diffeomorphism. Everything in this section, and the Frobenius theorem of Section 13.9.1 below, rests on it.

On differential forms (Section 13.6) the Lie derivative has a purely exterior expression:

The three operators \(\mathcal{L}\), \(\dd\) and \(i\) close on one another, and Equation (13.276) is only the first of the identities they satisfy. The second is the one that lets a computation move an interior product past a Lie derivative, which is what makes many arguments with forms one line long instead of a component calculation.

Distributions and the Frobenius theorem

A single vector field, by Theorem 13.125, threads the manifold with curves: through every point passes one integral curve, and the curves fill the manifold. The question of this subsection is what happens when a \(k\)-dimensional field of tangent planes is prescribed instead of a field of tangent lines. Does a \(k\)-dimensional surface pass through each point with the prescribed tangent plane everywhere? For \(k=1\) the answer is always yes; for \(k\ge2\) it is yes precisely when an algebraic condition on brackets holds, and that is the Frobenius theorem. It is the theorem behind the symplectic foliation of a Poisson manifold, behind the counting of gauge degrees of freedom generated by first-class constraints, and behind the very meaning of the word nonholonomic: a constraint is nonholonomic exactly when the distribution it defines fails the condition below.

The dual formulation is the one that appears whenever the planes are prescribed by equations rather than by spanning fields — a mechanical constraint written as a set of one-forms annihilating the admissible velocities is exactly that situation.

Killing vectors

A Killing vector is worth having because it gives a conserved quantity, and the conservation law — not the symmetry — is what a physical chapter actually uses. It is one line from Killing's equation.

Not every geodesic invariant comes from an isometry. The object that produces the rest is the symmetric-tensor generalisation of a Killing vector.

Metrics that saturate this bound are maximally symmetric (Section 13.13). Relaxing isometry to angle preservation defines the conformal Killing vectors, \(\mathcal{L}_{\xi}g = 2\sigma g\) with \(\sigma = \nabla_{\mu}\xi^{\mu}/D\); for flat space of signature \((p,q)\) with \(D > 2\) their maximal number is \((D+1)(D+2)/2\), the dimension of the conformal group \(\operatorname{C}_{p+q} = \SO(p+1,q+1)\) (Section 13.13.1).

Connections, parallel transport, and torsion

In Euclidean geometry a homogeneous (constant) vector field is one whose derivative with respect to position vanishes; its vectors are parallel and of equal length. How does this notion extend to an arbitrary manifold? The partial derivative fails: differentiating the vector transformation law \(V'^{\mu} = (\pp x'^{\mu}/\pp x^{\nu})V^{\nu}\) gives

whose second, inhomogeneous term spoils tensoriality whenever the coordinate change is nonlinear. The covariant derivative is the repair, and it is the concept on which the whole of theoretical physics leans.

Geometrically, torsion measures the failure of infinitesimal parallelograms to close: transporting \(\dd x_{(1)}\) along \(\dd x_{(2)}\) and vice versa, the two endpoints differ by \(T^{\lambda}{}_{\mu\nu}\,\dd x_{(1)}^{\mu}\dd x_{(2)}^{\nu}\). In Section 13.12 it becomes one of the two field strengths of the geometry.

The spin connection

Frame-indexed fields need their own connection: local Lorentz transformations Equation (13.265) are point-dependent, so \(\pp_{\mu}V^{a}\) is not covariant. The spin connection \(\omega_{\mu}{}^{a}{}_{b}\) enters as

and the requirement that frame and coordinate descriptions agree — the vielbein postulate, total covariant constancy of the vielbein — ties it to \(\Gamma\):

Metricity of \(\Gamma\) is then equivalent to \(\omega_{\mu ab} = -\omega_{\mu ba}\): the spin connection takes values in the Lorentz algebra \(\mathfrak{so}(p,q)\) (Lie Groups, Lie Algebras, and Fibre Bundles). Under a local Lorentz transformation it transforms inhomogeneously,

exactly like a gauge potential — the observation the Cartan formalism turns into machinery (Section 13.12).

Curvature

In the theory of curves a curve is characterized by its curvatures, built from first and second derivatives (Section 13.3.1); the curvature of a manifold must likewise live in the first and second derivatives of the metric, and vanish for Euclidean space. The correct object measures the failure of parallel transport around a closed loop, i.e. the non-commutativity of covariant derivatives.

The Cartan formalism

The results of Sections 13.8, 13.10 and 13.11 compress into two lines when written in the language of differential forms (Section 13.6), with all coordinate indices hidden. The dynamical variables are the vielbein one-forms \(e^{a}\) and the \(\mathfrak{so}(p,q)\)-valued connection one-form

and the Lorentz-covariant exterior derivative acts on frame-indexed forms as \(\mathrm{D}\alpha^{a} = \dd\alpha^{a} + \omega^{a}{}_{b}\wedge\alpha^{b}\) (one \(\omega\) term per frame index).

Maximally symmetric spaces and the symmetry ladder

The spaces admitting the full complement of \(D(D+1)/2\) Killing vectors (Proposition 13.140) are characterized by a curvature built from the metric alone, with one constant:

with \(T^{a} = 0\) and \(\ell\) a length (for \(\epsilon = 0\) the curvature vanishes and \(\ell\) drops out). In form language, \(R^{ab} = (\epsilon/\ell^{2})\,e^{a}\wedge e^{b}\).

The three cases realize the groups of Table 13.1:

• [Flat space (\(\epsilon = 0\)).] The manifold is \(\R^{p,q}\) with metric \(\eta_{\mu\nu}\); the Killing vectors are the \(D\) translations \(\pp_{\mu}\) (group \(\operatorname{T}_{p+q}\), abelian) and the \(D(D-1)/2\) pseudo-rotations \(x_{\mu}\pp_{\nu} - x_{\nu}\pp_{\mu}\) (group \(\operatorname{L}_{p+q} = \SO(p,q)\)), closing into the Poincaré group \(\operatorname{P}_{p+q} = \ISO(p,q)\), the semidirect product of the two.

• [Anti-de Sitter space (\(\epsilon = -1\)).] \(\operatorname{AdS}_{p+q}\) is the quadric

  \begin{equation}\tag{13.327} \eta_{AB}\,X^{A}X^{B} = -\ell^{2} \end{equation}

  in the flat \((p+q+1)\)-dimensional space of signature \((p,q+1)\), with embedding coordinates \(X^{m}\) and frame metric \(\eta_{AB}\) per the conventions of Section 13.1. The induced metric (Definition 13.119) satisfies Equation (13.318) with \(\epsilon = -1\), by Proposition 13.162 below. Every linear transformation of \(X^{m}\) preserving \(\eta_{AB}\) preserves the quadric and its induced metric, so \(\operatorname{Isom}(\operatorname{AdS}_{p+q}) \supseteq \SO(p,q+1)\); its dimension \((D+1)D/2\) saturates the Killing bound, so equality holds (connected component).

• [de Sitter space (\(\epsilon = +1\)).] Identically, \(\operatorname{dS}_{p+q}\) is the quadric \(\eta_{AB}X^{A}X^{B} = +\ell^{2}\) in signature \((p+1,q)\), with isometry group \(\SO(p+1,q)\).

The curvature of the two quadrics is computed once, for both signs at once, by the Gauss equation of Section 13.3.2 transcribed to a pseudo-Riemannian hypersurface of a flat space.

In the limit \(\ell \to \infty\) both quadrics flatten and their isometry groups contract to \(\operatorname{P}_{p+q}\) — the İnönü–Wigner contraction, treated algebraically in Lie Groups, Lie Algebras, and Fibre Bundles.

The conformal group

Relaxing isometry to angle preservation, \(f^{\ast}g = \Omega^{2}(x)\,g\), enlarges the symmetry. For flat space of signature \((p,q)\), \(D = p + q > 2\), the conformal Killing vectors are

for a total of \((D+1)(D+2)/2\) generators. That this list is complete is the content of the following proposition.

These close into the conformal group \(\operatorname{C}_{p+q} = \SO(p+1,q+1)\) of Table 13.1, realized linearly on the \((p+q+2)\)-dimensional flat space of signature \((p+1,q+1)\): flat \((p,q)\)-space is identified with rays of the null cone

(projective light-cone construction), on which \(\SO(p+1,q+1)\) acts by its defining representation, indices \(M,N\) and \(I,J\) per Section 13.1. The dimension matches: \(\dim\SO(p+1,q+1) = (D+1)(D+2)/2\).

The observed case

Physics instantiates this chapter at \(p + q = 3 + 1\). The kinematics of Part IV — Special Relativity is the flat row of Table 13.1; general relativity (Part V — General Relativity and Cosmology) is the torsion-free, Levi-Civita case of Sections 13.10, 13.11 and 13.12, tested by the experiments recorded there; and the accelerating expansion of the Universe (Evidence-Based Cosmology) makes the de Sitter row observationally relevant. Torsion, by contrast, is retained in this chapter as mathematics: no experiment to date requires \(T^{a} \neq 0\), and the physical parts of this treatise accordingly set it to zero — the scope rule of Epistemology and the Scientific Method applied to geometry itself.